## Abstract

A new type of two-dimensional photonic-crystal (PC) structure called annular PC composed of a dielectric-rod and a circular-air-hole array in a square or triangular lattice such that a dielectric rod is centered within each air hole is studied. The dielectric rods within the air holes greatly modify the dispersion diagram of the photonic crystal despite the fact that the percentage of volume occupied by the dielectric rods may be small (<12%). Increasing the radius of the inner-dielectric rod, starting from zero to a critical value, reduces the band gap and closes it completely as expected, because of the addition of more dielectric material inside the unit cell. Continuing to increase the radius of the rod above the critical value surprisingly creates another photonic band gap. Comparison of the dispersion diagrams of the new structure and the original lattice (circular air hole square/triangular array in dielectric background) reveals that the photonic band gap is considerably enhanced in size for both square and triangular lattice with the new structure. This approach preserves the symmetry of the structure and provides a complete photonic band gap away from the close-packed condition and at low normalized frequencies.

©2005 Optical Society of America

## 1. Introduction

Since the first proposal of multidimensional artificially periodic dielectric structures known as photonic crystals [1,2] (PC), tremendous progress has been made towards the realization of the main property of PCs, *viz*. the photonic band gap (PBG), which has enabled applications throughout the electromagnetic spectrum [3,4]. The PBG is a range of frequencies where the modes are evanescent and thus not allowed to propagate irrespective of the propagation direction and polarization type. Even though one can obtain the full benefit of the PC in three-dimensional structures, the challenge of manufacturing submicron-scale three-dimensional PCs at visible to near-infrared wavelengths has directed research efforts towards more easily fabricated two-dimensional PCs. The implementations of PCs include photonic band-edge lasers, highly reflective mirrors, superprisms, high-*Q* microcavities, low-threshold lasers, all-optical switches, optical transistors, and optical logic gates [5–16]. PBGs have been used to improve the gain and the far-field patterns of patch antennae in the microwave region [17,18]. Other applications in this region include high-impedance surfaces, compact uniplanar slow-wave lines, and broad-band filters [19-21]. We also recently proposed and carried out analysis of the PC waveguide structures for biochemical sensing [22,23].

There are some concerns related to making PC’s more functional and practical for deployment in real devices, even for the two-dimensional case. In particular, there are serious difficulties obtaining a wide PBG, for *all* polarizations, using simple, though widely used, geometries. For example, for many applications, one requires polarization-insensitive devices. This Letter directly addresses this problem by exploring a relatively simple geometry that is surprisingly effective in enabling the independent control of the PC properties associated with different polarizations. Namely, we consider a PC structure that combines desirable properties of two widely exploited slightly simpler geometries to attain this control. We also show that in some cases, the PBG can exceed the PBG’s expected from the two PC geometries that motivate our design.

Specifically, square and triangular arrays of circular air holes in a dielectric background and that of circular dielectric rods in an air background are the most commonly found PCs in the literature. The former (air holes) provides an absolute PBG near close-packed conditions. The latter (rods) usually does not provide a complete PBG, but there are ways as mentioned below to obtain a PBG even in this case. To have a PBG and to increase its size for all polarizations, to obtain polarization independent PBGs, or to engineer only TE or TM band gaps are all important aims. Employing an anisotropic material, reducing the structure symmetry, inclusion of metallic components, structural deformation, and composite structures are among the approaches to engineer the PBG that have been explored, and in these cases the PBG can be enhanced [24–29]. Each of these approaches, however, faces different problems. For example, metals are very lossy especially in the optical region. Therefore, very small metallic inclusions have been proposed. Structural deformation raises manufacturing concerns as the pattern of the lattice becomes complex. It is difficult to find strongly anisotropic materials. We are aware of no simple rules that can be drawn upon, such as which symmetry breaking structure gives larger PBG by lifting the degeneracies at the high symmetry points. The reduction of the symmetry typically also enlarges the high-order PBGs while reducing the lowest-order PBG. Finally, if a larger PBG is achieved at very close-packed conditions, then fabrication of the PC will be challenging due to the resulting very thin veins.

## 2. The structure

It is well known that a PC of isolated high-dielectric regions tends to support TM band gaps, whereas a connected lattice typically provides TE band gaps. One can illustrate this by referring to gap map for the specific type of PCs (square- and triangular-lattice dielectric rods in air and square- and triangular-lattice holes in dielectric background) [3]. The idea is to have a lattice that leverages off the strengths of both isolated and connected regions. As a result, there may be complete PBG.

The structure we consider was motivated by the desire to control independently, to the maximum extent possible, the size and the location of the TE and the TM band gaps. The convention for the TE and TM polarization is taken as the electric field is parallel to plane for the previous one and the magnetic field is parallel to plane for the latter one. The PBGs for both polarizations can be matched in frequency by selecting the proper structural parameters such as the radii of the rods and holes, and the refractive index of the various components of the structure.

The schematic diagrams of the structures are shown in Figs. 1(a) and 1(b). The dielectric rods of radius *r*
_{2} and permittivity *ε*
_{r2} are inserted in the middle of the holes with radius *r*
_{1} and permittivity *ε*
_{r1} (*r*
_{1}>*r*
_{2}). The new structure can be thought of as the combination of the dielectric rods in the low refractive index background and the holes in the high dielectric background. In general, *ε*
_{r1} and *ε*
_{r2} can be taken different; however, we assume here *ε*
_{r1} = *ε*
_{r2} = *ε _{r}*. The region where

*r*

_{2}<

*r*<

*r*

_{1}is denoted

*air*(

*ε*

_{0}) while other regions are denoted

*dielectric*(

*ε*);

_{r}*a*is the PC lattice constant.

From the PBG maps of square- and triangular-lattice rods in an air and square- and triangular-lattice holes in a dielectric background, we can see that the rods of relatively small radius tend to support substantial TM gaps while the holes of relatively large radius tend to support substantial TE gaps. The aim is to merge these two desirable regions of the disparate PBG maps by varying the radii of the rods and holes. As the rod radius increases, the band gap moves to lower frequencies for the TM polarization. On the other hand, the band-gap frequencies move to higher values as the radius of the holes increases. Thus, to have substantial overlap of both the TE and the TM gaps, one needs to increase (reduce) the radii of the dielectric rods while that of holes are decreased (increased).

## 3. Results and discussion

The photonic band diagrams are calculated by the planewave expansion method [30–32]. The computational error was estimated to be less than 2 % for the frequencies around the PBG. We first investigated the PBG variation as both the dielectric-rod radius *r*
_{2} and the air-hole radius *r*
_{1} are varied for the high-dielectric case *ε* = 13 (GaAs). It is obvious that when *r*
_{2} equals zero, the usual square and triangular lattice air-hole PCs are attained. One would expect that the PBG decreases with the addition of more dielectric material inside the holes by increasing *r*
_{2} starting from zero. It can be seen from Figs. 2(a) and (b) that the PBG closes completely at first as expected. However, another PBG appears in the dispersion diagram if *r*
_{2} continues to increase. For the square-lattice PC, as *r*
_{1} decreases the first PBG also decreases monotonically with smaller *r*
_{2} values. The second PBG is larger than the first one and the *r*
_{2} values corresponding to the peaks of the second PBG move to the higher values. On the other hand, the first PBG of the triangular lattice first increases and then begins to decrease as *r*
_{1} decreases and *r*
_{2} increases. The first PBG closes and a second one appears. As *r*
_{1} decreases, the second PBG first increases, reaches a maximum, and then decreases while *r*
_{2} values corresponding to the peaks of the second PBG move to higher values monotonically again. These observations based on Fig. 2 confirm our aim to overlap the PBG of TE and TM polarizations via increasing (reducing) the radii of dielectric rods while decreasing (increasing) that of holes.

Figures 3(a) and (b) show the dispersion diagrams of a triangular lattice of the circular holes with *r*
_{1} = 0.47*a* and the dielectric rods in the middle with two values of *r*
_{2} = 0.02*a* and *r*
_{2} = 0.14*a*. The two *r*
_{2} values were selected to show the effect of the first
PBG reduction in size and to indicate the opening of the second PBG. The insets are the first Brillouin zone of the triangular lattice. The dotted lines represent the TE modes and the solid lines represent the TM modes. The PBG is indicated as the shaded region that appears between the first TE and the first TM band gaps. The small inclusion of the dielectric rods reduces the width of the PBG as expected, at first up to a threshold value of around (*r*
_{2} = 0.09*a*), because the upper TM band (dielectric band) shifts downward. Meanwhile, the bands below the PBG remain similar compared to the bands above the PBG that moves towards the lower frequencies. However, increasing *r*
_{2} beyond the threshold value brings another PBG. The air band becomes the dielectric band for the second PBG in the dispersion diagram. Its size is comparable to the first PBG where gap width to mid-gap ratios are as follows for the two cases Δ*ω*
_{max}/*ω*
_{0} = 15.40%, *r*
_{1} = 0.47*a*, and *r*
_{2}= 0.02*a*; Δ*ω*
_{max}/*ω*
_{0} = 10.00%, *r*
_{1} = 0.47*a*, and *r*
_{2} = 0.14*a*. The PBG of the TE modes is reduced in size but that of the TM modes is increased in size. One astonishing result is that the upper edge of the TM band becomes flat even though other bands stay similar.

Similarly, Figs. 4(a) and (b) show the dispersion diagrams of a square lattice circular holes with *r*
_{1} = 0.49*a* and the dielectric rods in the middle with *r*
_{2} = 0.02*a* and *r*
_{2} = 0.11*a*. The reason for the selection of the *r*
_{2} values is the same as above and the insets show the first Brillouin zone of the square lattice. The dotted lines represent TE modes and the solid lines represent TM modes. The PBG is indicated as the shaded region that appears between the first TE and the second TM band gaps. The PBG behaves similarly to the previous case. The upper band edge of the second TM mode moves to lower frequencies swapping positions with the original PBG. It closes the PBG around a threshold value of around *r*
_{2} = 0.08*a* and becomes the dielectric band for the second PBG. The second PBG is wider than the first PBG for the square lattice: for the first case, Δ*ω*
_{max}/*ω*
_{0} =4.39%, *r*
_{1} = 0.49*a*, and *r*
_{2} = 0.02*a*; for the second case Δ*ω*
_{max}/*ω*
_{0} = 6.89%, *r*
_{1} = 0.49*a*, and *r*
_{2} = 0.11*a*. It is worth noting that the same lattice structure, either square or triangular, shows two PBGs in the same frequency region; one remains open only for small values of *r*
_{2} and the other one appears as *r*
_{2} increases beyond a critical value.

Inclusion of the dielectric rods within the structure shifts one of the TM band down. Changes to the inner structure (radius *r*
_{2}, permittivity *ε*
_{r2}) affect the TM modes to a greater degree. On the other hand, variation of the matrix parameters such as *r*
_{1} and *ε*
_{r1} have more effect for TE polarizations. Since the air band is sensitive to the dielectric material added, it swaps the PBG region, first closing it and then becomes the dielectric band for the second PBG. In Fig. 5 images represent the electric field at M point (band number = 4) for low and high dielectric inclusions. The nature of the 4^{th} band changes from air band to dielectric band. In Fig. 5(a), the field is mostly distributed through the veins and since the inner rod radius is small there is small field confined in the middle of the holes (higher frequency). However, for Fig. 5(b), the field is very well confined within the rods and reduces the frequency of the 4^{th} band (lower frequency). One important conclusion that can be drawn from Fig. 5 is that the tunability of the TM modes (n=4 for square lattice and n=3 for triangular lattice) is quite large compared to the TE modes, which stay relatively unchanged. The movie shows the same electric field variations as the inner radius changes between 0.06*a* and 0. 12*a*. From these plots and movie, one can conclude the above mentioned statement such that as the amount of inserted dielectric material increases field stays confined and its frequency decreases. As a result, the air band of TM modes becomes the dielectric band of the same modes.

We also looked at the case where the dielectric constant is *ε* = 16, as in germanium (Ge). An important feature is obtained. Though both the first PBG where *r*
_{2} = 0 and the second PBG where *r*
_{2} is greater than the critical value are enhanced with the high permittivity, the enhancement of the second PBG is almost doubled for the square and triangular lattice compared to the case of ε = 13 and the first PBG only increases slightly in size (Square lattice: the first PBG, Δ*ω*
_{max}/*ω*
_{0} = 4.83% to 6.50%, *r*
_{1} = 0.49*a*, *r*
_{2} = 0 and the second PBG, Δ*ω*
_{max}/*ω*
_{0} = 6.89% to 9.69%, *r*
_{1} = 0.49*a*, *r*
_{2}=0.11*a*. Triangular lattice: the first PBG, Δ*ω*
_{max}/*ω*
_{0} = 15.90% to 18.21%, *r*
_{1} = 0.47*a*, *r*
_{2} = 0 and the second PBG, Δ*ω*
_{max}/*ω*
_{0} = 10.00% to 18.34%, *r*
_{1} = 0.47*a*, *r*
_{2} = 0.14*a*). The first PBG is determined by the TM modes and the second PBG is determined from below by the TM mode and from above by the TE mode for the triangular lattice. Therefore, increasing the refractive index brings the first PBG to lower frequencies shifting the TM band pairs. However, for the second PBG, since the upper edge is TE and lower edge is TM, the TM mode shifts more than the TE toward the lower frequencies. Therefore, the increment of the second PBG as a result of the refractive-index change is larger. For the square lattice, the first PBG is determined by the TE mode from below and the TM mode from above. The second PBG, on the other hand, is bounded by TE mode. Both PBGs move to lower frequencies and due to differences between TE and TM modes response to the refractive index increment wider PBG was obtained.

There are two conditions that are important to achieve a PBG for all polarizations. One is to confine strongly the lowest order mode (dielectric band) in the dielectric region hence to lower the frequency and push the higher order modes in the air region hence increase the frequency. The other condition is related to either the radii of the dielectric rods or the amount of the dielectric background. The dielectric medium should have enough area to support primarily the lowest band and to be small enough not to be able to support higher order modes. The electromagnetic boundary conditions help elucidate why the TM modes are favored by the isolated dielectric rods and the TE modes are in the connected regions. Due to the boundary condition on the normal component of the electric flux density in the absence of charge density, the electric field is mostly in the air region for the dielectric rods surrounded by air. Thus the first condition is not satisfied for TE case. Consequently, there is no band gap. For the TM mode since the tangential component of the electric field must be continuous at the boundary, the field can be confined in the dielectric medium more strongly than in the TE case. The second condition can be achieved by having rod radius *r*
_{2} around 0.2*a* such that the higher order modes have nodes in the dielectric medium. For the holes in the dielectric background the scenario is different. Since one needs to have less dielectric material therefore radius of the hole should be fairly large. As a result, the connectivity helps TE modes to have a PBG. At some air-hole radius *r*
_{1} the lattice also looks like dielectric rods in air giving rise to a band gap for the TM polarization. Since the new structure (annular PC) has dielectric rods inside the air holes the rods are sensitive to TM polarization more than the dielectric background which plays similar role for TE polarization.

Since the symmetry of the lattice is preserved by the inclusion of the dielectric rods, the degeneracies at the symmetry points (Γ, M, and K) are unchanged. The structure is simple and the PBG is obtained at well less than close-packed conditions. In addition, the PBG appears at low frequencies, making it less susceptible to the disorder.

With the usual lattice geometry one can have only the radius of air holes and the background of the dielectric material to change the dispersion properties of the PC. On the other hand, annular PC’s provides extra two more variables due to inner rod (radius and refractive index) that can be utilized to alter the properties of the crystal such as to implement the tunability. This is another advantage of the annular PC. It may find other advantages especially when implemented in applications such as super-prism or self-collimations, switches, and splitters.

To show that the inclusion of dielectric rods (annular PC) is not the same as simply increasing the filling factor inside the holes, we have made several comparisons. Consider a unit triangular lattice annular PC (**r**_{1} = 0.15*a*, *r*_{2} = 0.45*a*, ** ε** = 13) in the four configurations: (i) dielectric rod (

*r*

_{1}= 0.15

*a*,

**=13) in the low index medium (**

*ε***= 4.468), (ii) the high index background (**

*ε***=13) filled with low index material (**

*ε*

**r**_{2}= 0.45

*a*,

**= 2.333), (iii) air holes in the high index medium (**

*ε*

*r*_{3}= 0.424

*a*,

**=13), and finally (iv) high index dielectric rods in air (**

*ε*

*r*_{4}= 0.309

*a*,

**=13). When we consider the dispersion diagrams of these configurations, there is no complete PBG for cases (i), (ii) and (iv), while there is PBG only for case (iii), which is almost half of the gap of the annular lattice structure. Thus, the approach presented in this work (annular PC) is clearly different than simply increasing the filling factor within the unit cell.**

*ε*The story is different when the dielectric contrast is low because it is difficult to achieve a complete PBG at low refractive-index values. At low dielectric values (*ε _{r}* = 4) the air hole square lattice has TE and TM gaps at close-packed conditions. Besides, the TE gap is very small. However, the new structure has a fairly high TE band gap as shown in Fig. 6(a) for the values of

*r*

_{2}and

*r*

_{1}that are likely to ease fabrication. For the air-hole triangular lattice, the TE band gap is available for very low dielectric values with large filling ratio. However, there is no TM gap for the low dielectric contrast case well away from the close-packed condition. On the other hand, the new structure has a very large TM band gap for a large range of

*r*

_{2}and

*r*

_{1}as shown in Fig. 6(b).

It has been shown that a larger PBG can be obtained as the lattice symmetry and the scatterer shape (e.g., circular or square) are the same i.e., square rods or holes in a square lattice and circular rods or holes in a triangular lattice [33]. It may thus be possible to enhance the PBG of the square lattice even more with the square rods inserted inside of square holes instead of circular rods inserted inside of circular holes.

The analysis is carried out for the ideal 2D case assuming infinite thickness in the z direction. We also presented the work with unperturbed annular PC. However, annular PC should have finite thicknesses for device applications. Since the confinement in the vertical direction relies on index guiding in this case, out-of-plane scattering losses become critical [34,35].

One may envision our annular PC structure either as a very thin slab supported by a low index substrate, such as silicon-on-insulator and operating below the light-line with excited Bloch modes, or as a photonic heterostructure, where the core is sandwiched between cladding layers on both sides (three-layer structure). In the latter case, the scattering loss is roughly proportional to the index contrast between the layers. Therefore, the index contrast should be low with deeply etched holes to reduce the scattering losses. In fact, increasing the depth of the cladding layers with small index contrast differences is an approximation to the ideal 2D PC. The origin of the out-of plane scattering losses is due to the air holes where there is no local guiding in the vertical direction. The radiation losses increase as the air fraction increases. Since the inner dielectric rods in the center of the each air hole reduce the distance that the light travels in air we think that annular PC will reduce that loss. Other losses due to manufacturing imperfections or structural disorders may degrade the performance of the
annular PC compared to usual PC as the former has one more sidewall within each unit cell. However, strengthen the arguments above, further work is needed. For example, analysis including the out-of plane losses (adding an effective imaginary * ε* within the air holes) and the finite thickness of the structure by the effective index method may be needed to provide better answers to the above questions.

## 4. Conclusion

In conclusion, we have analyzed the proposed structure, a two-dimensional annular photonic-crystal composed of a dielectric-rod and a circular-air-hole array in a square or triangular lattice such that the dielectric rod is centered within each air hole, in two different cases where the dielectric contrast is very high and very low. The inner dielectric rod reduces the PBG width and closes completely as expected. On the other hand, increasing the radius of the rod after the closure of the PBG gives rise to a second PBG. In some cases, this second PBG is larger than the PBG that one can get from the original PC lattice (circular and square lattice air hole array in dielectric background). Partial PBGs were obtained for certain polarizations for the low-contrast dielectric case. The ability to control TE and TM band gap with some degree of independence may vastly broaden the interest in two-dimensional PC structures.

## Acknowledgments

This work was supported in part by the National Science Foundation through grants ECS 0523923 and DMR 0305524.

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